What is the main resolution in "The Purloined Letter" by Edgar Allan Poe?
I would say that there are two resolutions to Poe's detective story. The first is that (in Poe's opinion) imagination and feeling are far more valuable characteristics to possess than are logic and mathematical skills. Poe advances this resolution not only through Dupin's lengthy diatribe on poetry, but also through Dupin's (a self-described poet) being able to solve the case that the supposedly logical, well-trained police cannot solve.
Secondly, Poe suggests that humans often tend to overlook the obvious and instead rely on the newest science and technology (or method of thinking) when they could simply use common sense. Readers can recognize this theory through the police's meticulous search of the apartment which results in nothing versus the perpetrator's hiding the stolen letter with other letters out in the open which Dupin, of course, discovers right away.
The story is an excellent example of the battle of thought that existed in Poe's day between the Romantic authors (Poe) and the Rationalists.
You might want to clarify your use of the phrase "main resolution." Typically, a resolution is the final part of a work in which loose ends are tied up and the plot structure is resolved. However, "main resolution" appears to be more of an argument term referring to the primary statement in a person's case (argument).
On the surface, you could say that the detective, Dupin, locates the letter and turns it over to the police. This resolves the main plot of the story (an early predecessor of the detective genre). However, this is Edgar Allan Poe with whom we are dealing. Do you think that there might be more to the story than the superficial? See if you can think more deeply about it. Perhaps you have more information regarding your assignment than you have expressed in your question and this might help you get started.
In Poe's Story "The Purloined Letter" or should we say stolen letter the minister has been having a dalliance with a female. He is aware that a letter has been stolen from her chambers that could ruin his career. He hires someone to find the letter for him and to bring it to him so that he can destroy it and continue his political career. What the minister does not know is that the man entrusted to help him has lied. He has made a copy of the original letter and pretended it is the actual letter. This will allow the minister to go on about his business confident that the issue is resolved. He is unaware that the real letter is there waiting to expose him and destroy his political career